If `|cot x+ cosec x|=|cot x|+ |cosec x|, x in [0,2pi],` then complete set of values of x is : `(a).[0,pi]` `(b).(0, (pi)/(2)]` `(c).(0,(pi)/(2)]uu[(3pi)/(2), 2pi)` `(d).(pi, (3pi)/(2)]uu[(7pi)/(4), 2pi]`

Options

A. `[0,pi]`
B. `(0, (pi)/(2)]`
C. `(0,(pi)/(2)]uu[(3pi)/(2), 2pi)`
D. `(pi, (3pi)/(2)]uu[(7pi)/(4), 2pi]`

Answer

To solve the equation \( | \cot x + \csc x | = | \cot x | + | \csc x | \) for \( x \) in the interval \([0, 2\pi]\), we will analyze the conditions under which the equality holds true.

### Step 1: Understand the Absolute Value Condition
The equation \( |a + b| = |a| + |b| \) holds true if both \( a \) and \( b \) are either both positive or both negative. In our case, \( a = \cot x \) and \( b = \csc x \).

### Step 2: Analyze the Quadrants
1. **First Quadrant**:
- In the interval \( (0, \frac{\pi}{2}) \), both \( \cot x \) and \( \csc x \) are positive.
- Therefore, \( | \cot x + \csc x | = \cot x + \csc x \) and \( | \cot x | + | \csc x | = \cot x + \csc x \).
- The equality holds.

2. **Second Quadrant**:
- In the interval \( (\frac{\pi}{2}, \pi) \), \( \cot x \) is negative and \( \csc x \) is positive.
- Here, \( | \cot x + \csc x | \neq | \cot x | + | \csc x | \).
- The equality does not hold.

3. **Third Quadrant**:
- In the interval \( (\pi, \frac{3\pi}{2}) \), both \( \cot x \) and \( \csc x \) are negative.
- Therefore, \( | \cot x + \csc x | = -(\cot x + \csc x) \) and \( | \cot x | + | \csc x | = -\cot x - \csc x \).
- The equality holds.

4. **Fourth Quadrant**:
- In the interval \( (\frac{3\pi}{2}, 2\pi) \), \( \cot x \) is positive and \( \csc x \) is negative.
- Here, \( | \cot x + \csc x | \neq | \cot x | + | \csc x | \).
- The equality does not hold.

### Step 3: Combine the Intervals
From our analysis:
- The equality holds in the intervals \( (0, \frac{\pi}{2}) \) and \( (\frac{3\pi}{2}, 2\pi) \).

### Final Answer
Thus, the complete set of values of \( x \) that satisfy the given equation is:
\[ x \in \left(0, \frac{\pi}{2}\right) \cup \left(\frac{3\pi}{2}, 2\pi\right) \]

This corresponds to option **(c)**: \( (0, \frac{\pi}{2}] \cup [\frac{3\pi}{2}, 2\pi) \).